Objectives

The primary objective of this experiment is to introduce the student to the measurement and response of a dynamical system. The experiments will occur on a tandem-rotor helicopter model connected to a pinned, swivel arm with a counter weight. Optical shaft encoders sense the motions of the helicopter model. During the experiment, we will calibrate the thrust control (voltage) input to the DC motors that control the rotors. Then we will measure the response of the system, specifically elevation, to various thrust inputs. Finally, the system response will be compared to a dynamical model of the helicopter.

Background

Dynamical Systems

A dynamical system is one in which the effects of an action do not achieve their impact immediately; the dynamical system evolves with time. The time evolution of the system depends both on the time-history of the external input(s) to the system and the properties of the system itself. Dynamical systems exist in many fields and applications:

• When a pilot changes the throttle (thrust) setting, the vehicle’s velocity changes only gradually – but the lighter the vehicle, the faster it can reach the final velocity

• When a cook puts something in the oven, its temperature does not rise to the oven temperature instantly – and larger items heat more slowly than smaller dishes

• When a company lowers the price of a product, the impact on the product’s sales (and the company’s stock) can take months to evolve – and that one input is not the only thing that effects the revenues (or stock price).

This way of looking at a system is in contrast to static approaches. For example in the previous labs, you looked at the response of a system to an external input: strain response of a system to a given stress, or lift on a wing as a function of angle-of-attack and wind speed. In those instances, however, we did not consider any rate issues; we assumed there was a unique (unvarying) relationship between the inputs and outputs of the systems. This was because we ensured that the rate at which we applied the changes to the inputs (or how long we waited to acquire the data) was much slower (or longer) than how fast the system took to reach a steady-state, i.e., how long it took to stop changing.

A powerful way to understand or predict the behavior of a dynamical system is through mathematical models. The models include variables that describe the current state of the system, the state variables, and the things driving the system, i.e., the external inputs. For example, we can look at a point mass that is free to move, the state variables would be: position ( $\vec{x}$ ), velocity ( $\vec{v}$ ) and acceleration ( $\vec{a}$ ), with three components of each if the mass can move in three dimensions. As you learned in physics for simple 1-d motion, these variables are not independent, i.e., $a = dv/dt$ and $v = dx/dt$ .

As an example of how one develops a model, we begin with a simple mass m pushed by an external force and constrained to move in one dimension. In this case, the motion of the mass is governed by Newton’s Law, i.e.,

where the notation .. represents the second derivative with respect to time. Now consider what happens if we add a spring that anchors the mass to a wall. Some of the force applied to the mass may now have to compress or expand the spring. You probably also learned in physics that the spring force can be modeled by Hooke’s Law (Eq. 1),

We can also add another device, called a damper, that produces a force that always opposes the motion of the mass – and the opposing force is proportional to the velocity, i.e.,

or equivalently:

Equation [4] above is an example of a linear second order system, which take a generic form as follows:

2. Since this is a closed-form equation, we need only plug in these values and the value of time we care about to get the corresponding position (as opposed to having to solve it numerically)

1-DOF Helicopter Model

The 1 degree-of-freedom (1DOF) helicopter mechanism used in this experiment is shown in Figure 2. It consists of a base upon which an arm is mounted (see Figure 3). The arm carries the helicopter body at one end and a counterweight at the other. Two DC motors with rotors mounted on the helicopter body generate a force (which we will call thrust) based on the voltages applied to each of the motors. The portion of the thrust generated by the rotors that is in the vertical direction can cause the helicopter body to lift, because the arm is held by a pin-type connector. The purpose of the counterweight is to reduce the power requirements on the motors.

The whole arm can pitch, which allows the elevation of the model to change by angle measured by an optical encoder. Natively, it can also roll and travel (yaw) in an azimuthal direction around the central swivel shaft, but this experiment locks those axes so that the system is purely 1-degree-of-freedom. All electrical signals to and from the main swivel arm are transmitted via a slip-ring with eight contacts, thus eliminating the possibility of tangled wires and reducing the amount of friction and loading about the moving axes. In this experiment, the two rotors will be driven by a control system with a Matlab/Simulink™ interface.

Modeling the helicopter

For this lab we will utilize a two point-mass model for the helicopter, namely that of the thrust-generating equipment, and the counterweight. Each of these masses is located at vertical and horizontal displacements as viewed in the body reference frame. In addition to the gravitational forces acting on the masses, the propellers generate a total thrust force, a spring force acts on the angular displacement, and a damping force acts on rate of change of angular displacement. The helicopter has a mass moment of inertia denoted J, that in our case will also be modeled via the two point mass model. This is summarized in Figure 5 below, with the variable definitions as follows:

In this lab we will measure the physical variables directly (lengths and masses), but the spring and mass constant can be notoriously difficult to predict so will address this in alternative ways. We will also directly measure the propeller thrust generation with an experiment.

Optical Shaft Encoders

The motions to be measured in this experiment are all related to the rotational motion of a shaft. Thus the Quanser system employs shaft (or rotary) encoders to measure the pitch, travel and roll of the helicopter model. Generally, a shaft encoder converts an angular position to an analog or digital output. While there are a number of electro-mechanical approaches employed in encoders, the most common employ either magnetic or optical sensors to read the shaft motion, with the latter more common, especially in high resolution encoders. These types of systems have replaced older electronic potentiometer-based systems.

Optical shaft encoders typically rely on the rotation of an internal code disc that has a pattern of opaque lines or shapes imprinted on it. The disc is rotated in a beam of light, e.g., from a small LED, and the pattern of dark and light regions on the disc act as shutters, blocking and unblocking the light in systems based on transmission, or strongly and weakly reflecting the light (see Figure 6). For high resolution, the disc is divided into many segments, typically in concentric rings. Internal photodetectors sense the alternating intensity of the light and the encoder's electronics convert the pattern into an electrical output signal.

There are two basic types of encoders: absolute and relative (also called incremental). An absolute encoder returns an exact position, or an angle in a rotary encoder, relative to a fixed zero position. An incremental encoder provides information on the instantaneous movement of the device, which can be converted to speed, distance traveled or position relative to an arbitrary initial state. Absolute encoders are necessary if a particular angular position must be known or retained, independent of when the encoder was powered on. Most other applications can employ an incremental encoder. Absolute encoders provide a output with a unique code pattern that is derived from independent tracks on the encoder disc which correspond to individual photodetectors and represents each position. Incremental encoders are simpler and provide information about the instantaneous motion of a rotating shaft by determining how many and/or how fast the alternating disc patterns go by.

Procedure

Equipment familiarization

Take a tour of the equipment with the TAs, they will show you:

• The experimental hardware

• Motors+propellers, and an explanation of how they are controlled

• 3-DOF to 1-DOF conversion

• Optical Shaft Encoder (incremental, 4096 counts per rev)

• USB DAQ (data acquisition device)

• Power amplifier (USB outputs voltage but can’t deliver enough current as propeller load is so high, so the amplifier provides this)

• Electronic weighing scale and stand (designed to hold the experiment at a perfect level)

• The Simulink model--how it communicates with the hardware and how to control the motors+props

Helicopter leveling procedure

1. If necessary, move the base of the helicopter to the back table--the one closest to the window (this table is slightly above the table that is in front of it).

   1. Move the helicopter using ONLY the base plate. Lifting the system using any other parts of the setup may damage it!

2. Place the electronic scale and stand underneath the thrusters. The leveling stand already takes into account the difference in table height.

The digital scale is designed to hold the main body level in a perfectly level position and be located directly below the thrust line (this means we can evaluate thrust without having to make a correction for leverage). The grooves in its top plate are deliberately narrow to provide this alignment but this can cause some interference if alignment is not correct, leading to false forces being measured by the scale. To avoid this, follow these steps:

1. Zero the scale if necessary

2. Locate the propeller tray sides in the grooves by moving the scale, do not move the helicopter base

3. Look down from above and fine-position the propeller tray so it's not visually touching the sides

4. Gently lift the propeller tray to check for interference all the way out of the groove... if it interferes at any point a readjustment is necessary

Encoder zeroing procedure

The encoders used in this experiment are incremental encoders, meaning there isn't an absolute reference of any kind. Instead, the hardware keeps track of how many steps increments of the encoder it has seen since being initialized. This initialization occurs upon connecting to the hardware, not when it is run. Therefore, follow these steps to zero the encoders:

1. Follow the helicopter leveling procedure above

2. In Simulink, click ‘Run On Hardware > Monitor and Tune > Connect’’

Static measurements

1. Download and open the MS Excel data recording template - Save a copy of this to your groups data folder (e.g. A20-X). DO NOT SAVE CHANGES TO THE TEMPLATE FILE. This excel sheet will be used to note down all your measurements during the lab

2. Referring to Figure 5 above, measure the force required to hold the helicopter level:

   1. Zero the digital scale and set its units to grams

      • TIP: give the Zero button a short sharp press as vertically as you can

   2. Make sure the counterweight is installed in the end hole

   3. Place the propeller tray in the digital scale grooves, taking care to prevent interference

   4. 1. The scale reading may bounce around and/or drift a little bit. Note down the most consistent reading from the scale as your data point.

3. Using the digital scale and a ruler/tape measure, obtain the following measurements:

Determine the propeller thrust mapping

1. Remove the counterweight from the helicopter if it isn't already (this enables the thrust equipment weight to prevent the heli from taking off)

2. Follow the helicopter leveling procedure above

3. Re-zero the scale

4. Open MATLAB and navigate to the experiment folder D:\AE2610\Helicopter\[Semester Year]

5. Open the Simulink file Helicopter.mdl

6. Once loaded, ensure the voltage is set to 0

7. Connect to the hardware by clicking ‘Run On Hardware > Monitor and Tune > Connect’

   • If you get a build error, click on ‘Run On Hardware > Monitor and Tune > Build For Monitoring’

8. Once connected, turn on the power amplifier

9. Click Start

10. Once running, command a voltage of 0V and note down the voltage and measured thrust (in grams) from the digital scale in Excel

    1. As before, the scale reading may bounce around and/or drift a little bit. Note down the most consistent reading from the scale as your data point.

11. Command a voltage of 1V and note down the voltage and measured thrust (in grams) from the digital scale in Excel

12. Command a voltage of 0V and wait for the propellers to stop. Re-zero the scale

13. Repeat steps 11 and 12 for a series of voltages from 2-10V in 1V increments, making sure to stop the propellers and zero the scale between each incremental measurement

14. Once finished, set the voltage to 0 and click Stop

15. Observe the Excel plot to check it makes sense (the TAs will help you here)

16. Re-take data points if your group deems it is necessary

Gather data to analyze linearized system response

1. Re-mount the counterweight in the end hole, with the top face roughly horizontal.

2. Using your data from Excel and the net force measurement from earlier, estimate the voltage required to hold the helicopter level. Write down this predicted voltage in your Excel file.

3. Follow the helicopter leveling procedure above.

4. Follow the encoder zeroing procedure above.

5. Making sure that the propellers do not slam onto the desk (have a member of your group hold the counterweight), move the electronic weighing scale and stand away from the model so that the model is free to pitch.

6. Slide the helicopter straight forward so that the propellers are well clear of the desk.

7. Ensure your group members are a good distance away from the propellers such that you do not interfere with them.

8. Input your predicted voltage.

9. Click Start.

10. Help the Heli find its natural settling angle by damping the oscillations. Do this by deadening the motion of the counterweight, DO NOT PUT YOUR FINGERS NEAR THE PROPELLERS!!! Use the Simulink Pitch scope to help you.

11. Adjust the voltage while the code is running until the Heli is at zero (it will be bobbling around a little at this point so you can use your judgement when it’s there). Use the Simulink pitch scope to help you here.

12. Write down the actual voltage at which your Heli is at zero in your Excel file. If there is a discrepancy, start thinking of reasons why this may be.

13. Once in position, sharply but gently push the counterweight either up or down to give the Heli a short impulse so that it oscillates roughly +/- 10 degrees. If you massively over- or under-shoot, this is not a problem but we need the logged data to be clean so re-level and dampen the oscillation, then repeat the impulse until you are happy. Again use the Simulink pitch scope to help you assess the quality of the data capture (the TAs will help you here).

14. Once the oscillations have been settled for a few seconds, slide the helicopter back in so that the propellers are well inside of the boundary of the table.

15. Repeat the impulse experiment in step 13 and discuss in real-time how, if at all, it varies from the previous experiment where the propellers were hanging over the table.

16. With the help of the TAs and the Pitch scope, determine if the impulse and subsequent data is good. If it is, set the voltage to 0 and click Stop. Catch the helicopter (by its counterweight) before it hits the table. If not, repeat the steps above until a good data set is obtained.

17. Verify the logged data:

    1. In the MATLAB Command Line, type: figure, plot(time, theta)

    2. Inspect the plot with the help of the TA to ensure it matches what you expect

18. If the data is satisfactory, right click in the MATLAB workspace and hit Save. Give the file an appropriate name. If the data is not satisfactory, repeat the experiment as needed.

Gather data to analyze nonlinear system response

1. Follow the helicopter leveling procedure above

2. Follow the encoder zeroing procedure above

3. Remove the digital scale and gently allow the propeller tray to settle in its natural position by holding and slowly releasing the counterweight (you may help it reach this position)

4. Click Start once it has fully settled

5. Push and hold the counterweight up so that the prop tray is firmly on the desk whilst being level

6. Cleanly release the counterweight so that you don't impart any external forces after letting go

7. Observe the oscillations on the Simulink pitch scope until the helicopter comes to its natural resting position again

8. With the help of the TAs, determine if the release and subsequent data is good. If it is, click Stop and catch the helicopter (by its counterweight) before it hits the table. If not, repeat the steps above until a good data set is obtained.

9. Verify the logged data:

   1. In the MATLAB Command Line, type: figure, plot(time, theta)

   2. Inspect the plot with the help of the TA to ensure it matches what you expect

10. If the data is satisfactory, right click in the MATLAB workspace and hit Save. Give the file an appropriate name. If the data is not satisfactory, repeat the experiment as needed.

Gather data to predict settling angle at different input voltages

1. Follow the helicopter leveling procedure above

2. Follow the encoder zeroing procedure above

3. Remove the digital scale and slide the helicopter straight forward so that the propellers are well clear of the desk (exactly as you did for the linearized system response experiment)

4. Gently allow the propeller tray to settle in its natural position (you may help it reach this position)

5. Set the Voltage to 0

6. Click Start

7. For the following voltages {0, 2.5, 5, 7.5, 10}, set the input voltage in Simulink and help the helicopter settle at whatever angle it wants to. Record the settling angle to voltage mapping in your Excel file.

   1. You may speed up this process by helping the helicopter settle at whatever angle it wants to settle at. Don't wait for the helicopter to naturally settle.

8. If your data looks good, set the voltage to 0 and click Stop.

If you have time...

1. Discuss with the TAs any other unmodeled phenomena that could be included in the model if we wanted

2. Do any other tests on the heli that you might find interesting

3. Get started on the data reduction while you have 1:1 time with a TA

Data To Be Taken

1. See Excel spreadsheet (note there are multiple tabs!!!)

2. One .mat file containing theta v. time for the linear response (with props spinning)

3. One .mat file containing theta v. time for the natural non-linear response

Data Reduction and Analysis

1. For the linear theta v. time response data - extract the time window of interest:

   • The start point should be the first time after the impulse has been applied that theta is zero and increasing (aka it should look like the start of a sinusoid with no phase shift)

   • The end point should be some point around when the oscillations have about settled (there will inevitably be some motion so do your best)

2. For the nonlinear theta v. time natural response data - extract the time window of interest:

   • The start point should be as soon as possible after the helicopter has been released

   • The end point should be some point around when the oscillations have about settled (there will inevitably be some motion so do your best)

3. For the voltage v. thrust mapping - determine a good fitting function that will enable you to plug in any value of voltage and get out a good estimate of thrust

Simulation Procedure

1. Modeling preliminaries

   2. Again using the two point-mass model, develop an equation that predicts pitch moment of inertia about the pivot as a function of the masses and their locations with respect to the pivot.

2. Linear modeling

   1. Using the nonlinear equation you derived in Step 1.1 above, develop a linearized form of it:

      2. Assume we manually set the thrust term such that the bias term is cancelled out.
   2. By comparing terms of your linear equation with those in Equation 4 in the lab manual, develop equations for damping ratio and natural frequency in terms of the model parameters.

   5. Again use Equation 5 to analyze a linear response, but this time use fresh values for natural frequency and damping ratio, instead manually adjusting them until you get the best possible fit you can. Again use the same time window from your reduced linear data.

3. Nonlinear modeling

   1. Using the nonlinear model you developed in Step 1.1 above, convert it to an anonymous function in MATLAB.

   3. Propagate another instance of the nonlinear model as you did in the previous step, but this time you have free license to modify any parameters to get the best fit possible.

4. Static balance analysis

   2. Again using your nonlinear equation from Step 1.1, develop an equation that predicts the natural settling angle (aka zero motion) as a function of input thrust.

   3. Using the data you acquired in the voltage v. angle mapping experiment, and the thrust v. voltage fitting you performed in data reduction, solve the previous equation to predict settling angle for the voltages you tested. Note that this equation will be nonlinear so not trivial to solve; you may use whatever method you deem appropriate such as Excel What If, MATLAB solver, trial and error, graphical, small angle approximation, etc.

Results Needed For DATA Report

1. Derivations of your following equations, showing all your steps:

   1. Nonlinear equation of motion

   2. Linearized equation of motion

   3. Pitch moment of inertia equation

   4. Damping ratio and natural frequency in terms of model parameters

   5. Predicted natural settling angle from thrust

2. Tables containing:

   1. All the dimensional/mass measurements you took in the lab

   2. Both empirically-obtained and model parameter-predicted damping ratio and natural frequency

   3. Predicted and measured settling angle by voltage input

   4. Predicted steady level hover thrust and measured leveling force from the digital scale

3. A plot of:

   1. Thrust versus voltage with 2 overlaid series; one of the raw data points you gathered in the lab, another showing the fit curve you made. Also provide the equation of the fit curve.

   2. The linear impulse response with 3 overlaid series; measured, model-parameter-predicted, and manually adjusted.

   3. The nonlinear natural response with 3 overlaid series; measured, model-parameter predicted, and manually adjusted.

4. Discussion of:

   1. How well the empirically-obtained and model parameter-predicted damping ratio and natural frequency match up, including any thoughts on why any discrepancy may have arisen.

   2. How well the predicted and measured settling angle match up, including any thoughts on why any discrepancy may have arisen.

   3. How well the steady level hover thrust and measured leveling force match up, including any thoughts on why any discrepancy may have arisen.

   4. What fit curve you used for the thrust-voltage mapping fit and why you picked that one, plus some thoughts on how well it fits the measurements.

   5. How well the linear impulse response series match each other with thoughts on why any poorly-fitting series are that way

   6. How well the nonlinear natural response series match each other with thoughts on why any poorly-fitting series are that way

   7. Why you would use each of the modeling techniques covered in this lab, with particular consideration of: ease of implementation, current design stage, accuracy, etc.

Here you will find the lab manuals for all experiments that will take place this semester. Navigate to the appropriate experiment on the left-hand menu. You can export each experiment as a PDF if you want a copy for your records, or to print as a hard copy. Also, bookmark this page on your smartphone or tablet as well as your PC/laptop as a handy guide during your lab session.

Note that this semester is the first-time we are rolling out Gitbook to deliver the manuals, so please let your TA or Instructor know if there are any typos, or suggested improvements...

Best of luck for the semester!

Objective

The primary objective of these experiments is to familiarize you with digital data acquisition of time-varying signals. This lab covers concepts in frequency analysis of time varying signals and sampling theory. It also provides an introduction to digital data acquisition (DAQ) systems. You wil use a personal DAQ to sample signals produced by waveforms stored in mp3 formats and converted to analog electrical signals. You will explore issues in: sampling, including the Nyquist limit and aliasing; filtering and its use for noise reduction; and digitization errors. You will also use your DAQ and a microphone to record and analyze sounds in an experiment chosen by yourself.

Background

Most experimental measurements involve the dimension of time. Experimental data is acquired over the course of some time, and during this time the actual physical parameter of interest (the measurand) may change. This could be due to a transient, such as the stress induced in a material by a sudden impact, a periodic phenomena, like the bending and twist of a helicopter blade due to flutter, or random or chaotic fluctuations, like the turbulent velocity in a wind tunnel. Even when the measurand is nominally constant in time, other parameters that influence the measurement may vary, for example drifts in the measurement device. Thus, the experimenter is often interested in measuring a variable that can be described by the general function (or waveform),

(1)

Waveforms and Frequency Content

Fourier Series

One of the simplest time-dependent functions we encounter is the sine (or cosine*),

(2)

where $A$ is the amplitude, $\omega$ is the circular frequency (e.g., rad/s), $f$is the cyclic frequency (e.g., cycles/s, Hertz or $s^{-1}$), and $\phi$ is the phase, which represents the time-shift of the sine-wave from some reference time that defines $t = 0$. Such a function is often denoted as a simple harmonic waveform.

More general periodic waveforms, which repeat themselves with a period T and thus have a frequency $f = 1/T$, can be written as a linear combination of simple harmonic modes. There is the basic, fundamental mode (with frequency$f$), and harmonics of the fundamental mode, with integer multiples of its frequency ($2f$, $3f$, …). For example, we could describe the vibrations of a tuning fork or the acoustic oscillations in a pipe this way. Mathematically, this linear combination of modes is expressed as a Fourier series expansion,

(3)

where $nf$ represents the frequency of the $n^{th}$ mode ($n = 1$ for the fundamental, $n = 2$ for the first harmonic, etc.), $a_0$ represents the steady component of the waveform, and the $a_n$, $b_n$ are the harmonic coefficients (or amplitudes) of each mode. The steady amplitude, $a_0$, is often called the DC component of the waveform, in reference to classical electrical power systems, which are either Direct Current (steady) or Alternating Current (sinusoidal with a zero average).

For example, Figure 1 shows a simple waveform composed of two frequencies, a fundamental mode at 50 Hz and its 9th harmonic. Thus the complete waveform is repeated every 20 ms (period=1/fundamental frequency =1/50 s). The waveform shown in the figure also has a DC component. In other words, the signal has a nonzero value when averaged over its period. In general, we can write the DC amplitude as

(4)

The other coefficients of the Fourier expansion are given by

(5)

and they can be combined into a complex number (since, $e^{-ix}=\cos x-i\sin x$),

(6)

The power, $P$, contained in single mode is given by the square of the amplitude

(7)

and the phase $\phi$ (or phase angle) of a mode is given by

(8)

A second example that shows the ability of a combination of sine waves to create an arbitrary periodic function is shown in Fig. 2. Five sine waves and a DC component (see Fig. 3) were combined to create a function approaching a square wave. While the constructed function resembles a square wave, it is clear that more sine waves would be needed to produce a sharp square wave.

Fourier Transforms

The procedure outlined above for periodic functions can be extended to general functions, which are not necessarily periodic, by considering any arbitrary function to be periodic with an infinitely long period. This approach leads to the Fourier Transform. Given a function $v(t)$, its Fourier Transform $V(f)$ is a complex function defined by

(9)

in parallel to the complex Fourier function of equation (6). The function $V(f)$ represents the information given by $v(t)$ transformed from the time domain to the frequency domain. The transformation is nearly identical in the reverse direction, with simply a change in the phase (note the sign of the exponent), i.e.,

(10)

For example, Figure 4 graphically shows the Fourier transforms of various functions, including sine and cosine waves, a rectangle function (Π), a triangle function (Λ) and a constant, or DC, function. The sine, cosine and DC waveforms result in Fourier transforms that are nonzero at a single frequency**;

in other words, they contain information at only one frequency (the DC function, which does not change in time, is associated with a frequency of zero). The rectangle and triangle functions shown are not repeating like the sine waves; the functions shown here represent "single pulses." The Fourier transforms of these single rectangle and triangle functions result in $\text{sinc}$ and $\text{sinc}^2$ functions, where $\text{sinc}(f)=\sin(\pi f)/\pi f$ , which contains information at many frequencies, but with multiple frequency “peaks”.

Instead of looking at the Fourier transform, we often are interested in the power spectrum (or power spectral density, PSD) of a waveform. This represents the amount of power or energy in a region between $f$ and $f+df$. For real (noncomplex) functions $v(t)$, this is given by

(11)

where it is sufficient to consider only $0 < f < \infty$ since the PSD of a real function is symmetric about $f = 0$.** Thus the PSD of the rectangle function, $\Pi(x)$ as shown in Figure 4, is the square of its Fourier transform, or $\text{sinc}^2(f)$ (also shown in Figure 4).

Extensions of the Fourier Transform method have been developed for non-continuous functions, specifically for signals that have been discretely sampled by a computer, data acquisition system, or produced by digital means. These are generally known as Discrete Fourier Transforms (DFT), and a computationally efficient approach is known as the Fast Fourier Transform (FFT). These concepts are described in detail in References 2 and 4. The digital data acquisition system you will use employs these techniques to compute power (and phase) spectra.

An example PSD is shown in Fig. 5 for an 80 Hz sine wave based on a DFT. Unlike the Fourier Transform of the sine wave in Fig. 4, the PSD for a sine wave is no longer infinitely thin, i.e., the DFT shows power spread over a small range of frequencies around 80 Hz. This occurs because the DFT has a limited frequency resolution (it can only report values at discrete frequencies).

​Noise

Measured signals in ground experiments and flight tests (as well as other applications such as communications and controls) include noise from various sources. In the frequency domain, noise can have a very complicated structure. There are some simple noise models, however, that can be appropriate in many situations. For example, white noise has a flat power spectrum, meaning it has the same power at every frequency over some wide range. Another type of noise observed in many systems, including electronics, music and many biological systems, is called $\bm{1/f}$ noise (or pink noise). In this case, the power spectrum (again over some wide frequency range) scales as the inverse of the frequency, i.e., the power of the noise at each frequency is inversely proportional to the frequency. Both of these types of noise can be observed at the same time (as well as other noise types). For example, Fig. 6 shows a power spectrum (power spectral density, PSD vs frequency) with both $1/f$ and white noise components. Using these models, one can interpolate the noise at a frequency that has "real" signal, and estimate the signal-to-noise ratio (PSD of signal at a given frequency divided by the PSD of the estimated noise).

Sampling Theory and Aliasing

For most situations involving either computer generated or computer acquired data, the continuous function $v(t)$ is sampled at evenly spaced, discrete intervals in time, separated by an amount $\Delta t$. The sampling frequency (or data acquisition rate) is thus $f_s=1/\Delta t$.

For a given sampling rate, we might ask how accurately the discretely acquired data can reproduce the actual waveform being sampled. The answer depends on the frequency content of the waveform and a special frequency, called the Nyquist frequency $(f_N)$, which is half the sampling frequency, i.e., $f_N=f_s/2$. If the waveform contains no components above the Nyquist frequency, then the waveform can be completely determined by the sampled data (assuming no errors in the measurement).*** This is known as the Nyquist/Nyquist-Shannon Sampling Theorem.

As a simple example, consider a single sine wave. If we know we are dealing with a single frequency sine wave, it takes at least two measurements per period to determine its frequency, which means we must sample at twice the sine wave’s frequency. If we sample any slower, we actually infer a lower frequency than the actual frequency of the sine wave (you will see this in the lab). This process, by which information at a higher frequency shows up at a lower frequency is known as aliasing.

Aliasing occurs for any sampled waveform having components with frequencies above the sampling system’s Nyquist frequency, i.e., $(f>f_N)$. One way to remove this problem is to filter the data before it is sampled. This can be accomplished by a low pass filter, a filter that only passes frequencies below some cut-off frequency. One would set the cut-off at or below the Nyquist frequency. The high frequency information is thus removed before it can be aliased. In essence, the filter produces a bandwidth limited waveform.

Digital Data Acquisition Systems

Data will be acquired with a standalone digital data acquisition system (DAQ), LabJack™ T4, that communicates with your computer through a USB connection and using a LabVIEW™ software interface. Most DAQs can be connected to more than one input source; each signal (e.g., a voltage) is connected to one channel of the DAQ. A typical DAQ consists of a multiplexer, a sample-and-hold device, an amplifier, an analog-to-digital converter, a memory buffer, a microcontroller, and an interface to a computer (see Figure 7).

The multiplexer** ** (MUX) is a switch that connects one of a number of input channels (usually numbered starting at 0) to the sample-and-hold** ** (S/H). The input voltage on the channel switched by the MUX “charges up” the sample-and-hold during some time interval, which is a fraction of the sampling period (the time between samples). This circuit is then disconnected from the input voltage, and some of the stored charge is drained from it. The amount of charge leaving during this time is proportional to the original input voltage. The output of the S/H is amplified and then converted to a digital value by the analog-to-digital converter** ** (ADC). The digital result is then moved to the buffer memory, and communicated to the computer.

The digital value produced by the ADC (sometimes referred to as a “word” of data) depends not only on the input voltage, but also on the voltage range and number of bits of the ADC/amplifer system. The range is given by the minimum and maximum voltages that the ADC/amplifier can read (e.g., 0 and 5 V). The number of bits ($N$) in the ADC determines its digital dynamic range (= $2^N-1$). Thus the relation between the digitizer output and the voltage input is given by

(12)

where output has to be an integer value. As an example, for a 2.05 V input into a DAQ with a 0-10 V range, and an 8-bit digitizer (possible digital values of 0-255), the output value would be 52 (not 52.275). Any signal amplitude variations below the difference between two adjacent quantized levels are lost; this is known as the quantization error =$(maximum-minimum)/2^N$. In the example above, we can only say the input value was 2.039V$\pm$0.0196 V (assuming the example ADC rounds rather than truncates). One would normally choose an ADC with a number of bits sufficiently high that the quantization error is less than the dominant sources of error in the measurement. Other factors, though, may influence the choice of ADC bits, including cost and data storage requirements, both of which increase with the added number of bits.

Multiple signal inputs are recorded by using the MUX to cycle through each of the input channels at a rate that must be faster than the overall sampling rate (how often a given channel is read) times the number of input channels being read. In the sequential sampling system illustrated in Fig. 6 (and which is representative of the system you will be using), note that the channels are not read at exactly the same time. There is a time delay (skew) between when one channel and the next is read. The skew is determined by the maximum switching and reading rates of the MUX, S/H and ADC. This is illustrated in Fig. 8. Simultaneous data acquisition systems, which have negligible skew, typically employ multiple, synchronized S/H systems just upstream of the MUX (see Fig. 9).

In this lab, you control the data acquisition process through a software interface called a LabVIEW virtual instrument (VI). The VI creates a display on the computer screen that lets you think of the data acquisition system as a box with “knobs”, “dials”, and other displays. For this experiment, the VI allows you to control parameters such as the minimum and maximum voltages read by the DAQ, the sampling rate$(f_s)$, and the number of samples recorded.

Sampling/DAQ Terminology

The following terminology is commonly used in DAQ systems, and you should become familiar with these terms.

• Sample = a single measurement (i.e., at an "instant" in time) captured by the DAQ from one channel

• Sampling period = the time between two successive samples

• Sampling rate = 1/sampling period, with typical units of Samples/sec (S/s) or Hz

• Record = a group of successive samples acquired by the DAQ

• Record length = the number of samples in a record, typical units of Samples (S)

• Record duration = the time between the first and last sample in a record

Procedure

Mac Users

The software for this lab currently only works on Windows, and cannot be made to work for Mac (or any other OS). This leaves Mac users with the following options that we have tested and verified:

• Borrow a Windows machine from a peer, or at the Georgia Tech library

• Install and use Parallels

  • This works for both Intel and Apple chips.

  • It is not free but a student version costing $39.99/year is available

  • This is a "virtual machine" that creates an instance of Windows running within your Mac OS. As such, no reboot is required, and only 500Mb for the installation file is required.

  • When running, Parallels will take around 4Gb of RAM.

  • You will also need a copy of Windows to install on the partition; see here for instructions on how to get a free student copy of Windows.

  • Windows 10 Education (not the N version) is recommended since W11 requires a lot more memory.

• Use the free in-built Bootcamp software

  • This only works if you have the Intel chip.

  • This "dual boots" your machine which involves partitioning your drive with >64Gb of your drive's space permanently allocated to Windows. You will also need to reboot every time you want to switch between operating systems.

  • You will also need a copy of Windows to install on the partition; see here for instructions on how to get a free student copy of Windows.

  • Windows 10 Education (not the N version) is recommended since W11 requires a lot more memory.

• Install and use the free Oracle VirtualBox

  • This works only for Intel chips.

  • This is a "virtual machine" that creates an instance of Windows running within your Mac OS. As such, no reboot is required, and only ~125Mb for the installation file is required.

  • When running, VB will take at least 0.5Gb of RAM.

  • You will also need a copy of Windows to install on the partition; see here for instructions on how to get a free student copy of Windows.

  • Windows 10 Education (not the N version) is recommended since W11 requires a lot more memory.

Free student Windows license key

If you need a Windows download and a license key, Microsoft's Azure For Education program gives you access to Education versions of Windows OS for free. Head over to that link, sign in with your GT credentials, and download from there.

Borrowing from the library

If you don't have the ability to borrow a Windows machine from a peer, you will need to check out a loaner Windows laptop from the library. Follow this guidance:

• Library laptops can only be checked out for 4 hours at a time. However, they can be renewed at the library for another 4 hours once per day.

• GT-OIT has programmed the machines to fully reset to their nominal state if it is shut down or restarted. Therefore, ignore any instructions to restart the machine after software installation. Instead, select RESTART LATER.

• When you return the laptop to the library but immediately check out again to extend your time, try and check back out the same machine, else you will need to repeat the installation.

• Remember to send yourself any saved data before returning the laptop.

• Log in to the laptop using the default user account credentials supplied on the device, DO NOT use your GT credentials otherwise you won't have admin rights to install software

• Once you've acquired the laptop, follow the instructions below as normal.

Prior to Week 1

The following tasks should be accomplished prior to meeting for the first week of lab. The intention of this section is to ensure you have working software on your computer before coming to the lab. The hardware can only be tested if the software is installed so make sure to complete this step to prevent delays during the lab.

1. Software preparation:

   • Go to the LabJack software installation page and download the "Release" version for your operating system and screen size

   • Run the installer and follow all steps to install in your preferred location

   • Download and unzip the GTAE Simple DAQ installation files (found in the Software and Files section below)

   • Run the installer and follow all steps to install in your preferred location

   • Restart your machine

Week 1

The following tasks should be accomplished during the lab. The intention of this lab is to make sure you verify all items in your kit functions correctly and that you have a basic understanding of how they function.

1. Kit preparation:

   • Obtain a Digital Sampling kit from your TAs

   • Check that the contents of your Digital Sampling kit match the box label; if anything is missing, obtain any replacement parts from a TA. The kit should contain:

     • LabJack T4 DAQ

     • USB-A to USB-B Power Cable

     • LabJack Screwdriver

     • Miniature Eyeglass Screwdriver

     • Microphone

     • 3.5mm Audio Breakout Adapter

     • At least 3 male-male Jumper Wires

     • Female 3.5mm Audio Jack to Male USB-C Adapter

   • Register your LabJack T4 serial number with your TAs

2. Verify your LabJack T4 DAQ functions correctly:

   • Connect your LabJack T4 DAQ to your computer using the provided USB cable

   • Open LabJack's Kipling software that you previously installed, and connect to the T4 by clicking on the green USB button (refresh devices if it wasn't found)

     • !!! NOTE - If your search bar doesn't find it, navigate to Kipling under the LabJack folder after pressing the Windows Start icon !!!

   • Go to the Dashboard tab, where you will see a schematic of the T4 with live display of the inputs and output pins. Perform the following "loopback" tests to make sure your DAQ is functioning correctly:

     1. By default, AIN0 to AIN3 should be reading around 1.4V

     2. With one of the jumper wires and larger screwdriver from your kit, connect AIN0 to a nearby GND pin; the AIN0 voltage should now go to around 0V (the pin is being pulled to ground)

     3. Now connect AIN0 to VS (supply voltage); since VS is coming from your USB port you should be reading around 5.1V

     4. Now set DAC0 and DAC1 to 2 different voltages less than 5V (DAC means Digital-to-Analog-Converter i.e. these pins are analog voltage outputs); connect AIN0 to each of them, verifying that the measured voltages match the output voltages

   • Disconnect wire and close Kipling

3. Verify your peripherals function correctly:

   • Ensure your T4 is plugged in

   • Open GTAE Simple DAQ, choose a save data folder, and the application should now be running. !!! NOTE - Sometimes the SimpleDAQ will throw an error on startup or mid-acquisition. If this happens, restart the software and power cycle (unplug and plug back in) the DAQ !!!

   • If your monitor is clipping the edges of SimpleDAQ, try adjusting the display resolution and scaling (zoom) until it fits. In Windows this is done by right-clicking on the Desktop and selecting 'Display Settings'. If no options work for you, try plugging in an external monitor.

   • Turn on Autoscale for both axes of both plots; Time History and Power Spectrum. You should now see a real time plot of channel AIN0 with a voltage of around 1.4V plus or minus a small amount of noise

   • Test your 3.5mm audio breakout adapter:

     • Retrieve the breakout adapter and, with one jumper wire, connect either its L or R terminal to your DAQ's AIN0 port (note that you may need to open up the screw terminal on the DAQ before being able to insert the wire)

     • With another wire, connect the ground terminal (far right) to your DAQ's GND port (the one next to AIN0)

     • Ensure you have good wired connections by gently tugging on them to ensure they don't come loose. Be careful not to bend the pins as they can snap fairly easily.

     • Plug the audio breakout into your laptop/tablet/smartphone (any device that can play audio and access YouTube).

       • In Windows you will likely get a popup asking which device you just plugged in. Here you should select whichever option gives the best signal quality with low noise/distortion... this should be done iteratively and using your best judgement.

     • Ensure that your operating system is set to play over the headphone jack (i.e. not your bluetooth headphones or in-built speakers).

     • Set the YouTube video player to maximum volume and your device volume to roughly 50%. You may need to turn it up higher if your waveform has low amplitude.

     • Go to YouTube and find a "human hearing test" video such as this one

     • With the video playing, observe what's happening in GTAE Simple DAQ; you should see a sine wave of increasing frequency both in the time history and as a shifting spike in the power spectrum that should match that shown in the YouTube video (until around 11 kHz when it will start to diverge)

     • Adjust your device volume to observe the shifting amplitude of the signal

     • Once you have verified correct functionality after a minute or two (don't play the whole video), stop the video, unplug the audio adapter and remove the wires from the adapter (don't remove them from the DAQ)

   • Test your microphone:

     • !!! WARNING - the microphone has bare electronics on the back of the board which can result in damage if shorted with a metal tool or surface; be sure to handle carefully and away from metal when powered !!!

     • Retrieve the microphone and micro screwdriver from your kit

     • Connect the GND port of the microphone to the GND port on your DAQ which should already have a wire connected (note that you may need to open up the screw terminal on the microphone before being able to insert the wire)

     • Connect the OUT port of the microphone to the AIN0 port on your DAQ which should already have a wire connected

     • Connect the VCC port of the microphone to the VS port of the DAQ with a new wire

     • You should now see a rough waveform in GTAE Simple DAQ centered around roughly 2.5V. To stop the time history jumping around so much, turn off Y-Axis Autoscale and set the scale's maximum and minimum directly. Do this by double-clicking on the highest and lowest and numbers on the axis (6V and 0V might be a good start but adjust as necessary)

     • Test the microphone by generating some sounds of interest such as whistles, taps on the desk, talking, playing a tone from headphones, etc. Don't get the microphone too close to the noise source, it is quite sensitive. Bear in mind there will be lots of background noise so you may want to go somewhere quiet. !!! Note - the front of the mic is the black pad inside the metal can... point this towards your noise source !!!

     • Once you are satisfied the microphone is functioning correctly, remove all wires from all terminals.

   • !!! WARNING - the DAQ, and especially the microphone are fragile items. MAKE SURE they do not get tossed around or they will break and you will be unable to finish your lab !!!

4. Before leaving the lab, understand the objectives and resources for the next 3 weeks:

   1. Read the following steps in this manual that outline what you must have completed prior to each lab session. Check with your TAs if you are unsure of anything.

   2. A more in-depth explanation of GTAE Simple DAQ is given at its dedicated webpage, including a short video on how to use the software.

   3. The LabJack website is a great resource for understanding more about how the DAQ works, and what you can do with it.

Prior to Week 2

By Week 2 lab, at a minimum, you must complete the following tasks, but you are free to try other things to learn about digital data acquisition, sampling theory, and frequency content of signals. Keep notes on what you observe or find as you do each task and keep good, clean, legible, and organized notes. Answer any questions that are posed. All of your observations, notes, and answers to questions posed this will be checked by the TAs during Week 2 lab and graded for accuracy/effort!

You can work on this wherever and whenever you want (but before your regularly scheduled lab session). Also, you must work on these task by yourself. If you have trouble with the equipment or do not understand the required tasks, you can reach out to the TAs during office hours or using Piazza.

You will still attend your lab session at your regularly scheduled time, this week - bring your DAQ/computer system with you. You will be asked to demonstrate certain things to the TAs and answer some questions based on having done these tasks and learned the underlying concepts. If you can not do the required tasks or successfully answer the TAs questions, you can work during the lab time to work on the material, and be re-assessed by the TAs when you think you are ready. You have an unlimited number of attempts to pass the assessment, but only until the lab session ends. Your TA's can and WILL take points off if you come into Week 2 lab with nothing finished or it seems like very little effort was put into it. Furthermore, you will be required to finish everything during lab.

1. Setup DAQ system

   • Following the steps from Week 1, configure your DAQ and 3.5mm audio breakout adapter so that your laptop/tablet/PC/smartphone can capture the waveforms in the above YouTube video

   • !!! WARNING - Some devices (usually PCs/laptops) do not have good sound cards, resulting in significant distortion of the waveform. You will know this is happening if you don't recognize any square/triangle/ramp waveforms in the subsequent steps, and even the sine waves have significant distortion. If this occurs, switch to a device designed more for sound playing such as a smartphone or tablet (check with other members of your group or friends if you don't have one). You could also download the tracks and import them to an MP3 player. Please use the USB-C to 3.5mm adapter provided since this has historically yielded better results. If all else fails, contact a TA who will help you find an appropriate device !!!

   • In GTAE Simple DAQ, set the following settings:

     • Sampling rate = 25,000 S/s

     • Record length = 1,000 S

     • Autoscale = initially ON for all axes (you will need to turn this off to adjust the time scale to "zoom in" during waveform identification)

     • All other settings should be okay based on the defaults when you start the application

2. Use DAQ to perform waveform identification

   • In the Waveforms section below you will find 11 different audio tracks. Each audio track contains a different periodic signal. These signals include: single sine waves (at different frequencies), a sum of three sine waves (each at a different frequency), a product of two sine waves (e.g., sin(At) x sin(Bt), also known as amplitude modulation), a sine wave of a sine wave (e.g., sin(sin(At)), also known as frequency modulation), and periodic waveforms that are not sine waves: square waves, triangle waves, and ramps. Some tracks also have "noisy" versions of some of these waveforms.

   • For each track, observe the time plot and power spectrum (adjust the output/volume level as required to see the waveforms clearly)

     • Tip: With the waveform displayed as you like, you can toggle the Continuous/Hold switch to the Hold position so that the display just shows the last data captured (doesn't keep taking new samples)

       • Qualitatively identify the waveform on each track (e.g. sum of 3 sines)

       • For each track, write down the frequencies for each of the peaks you see in the power spectrum along with its respective power level

         • Tip: you can turn the autoscale off on the Power Spectrum x-axis and set the axes limits to make it easier to find the frequency(s)

3. Examine complex waveforms and interpret power spectra

   1. Play the track you identified as product of sines (amplitude modulation)

      • From the time plot, determine the period of the wave (time between peaks)

        • Tip: remember you can toggle the Continuous/Hold button if you want to freeze the display

      • Is the frequency of the waveform based on the period the same as any of the frequencies you wrote down from the power spectrum? If not, why?

        • Hint: the power spectrum shows the frequencies needed to produce that signal from a sum of sines!!!

   2. Play the track you identified as the triangle wave

      • From the time plot, again determine the period of the wave (time between peaks); how is it related to the frequencies you wrote down from the power spectrum?

   3. Alternate playing the triangle wave track and the square wave track

      • Compare the heights (power) of each peak in the power spectrum; which waveform has more power at high frequencies?

      • Think of any reasons why that waveform should contain more high frequency content based on its shape

4. Examine quantization error

   • Set your computer volume to max. Play the track with the square wave.

   • Make sure the settings are: Continuous, Sampling Rate = 10,000, Record Length = 1000, all axes have Autoscale=on except time plot, set maximum limit on time plot x-axis to 0.01 seconds. Turn off time plot y-axis autoscale.

   • Observe both the time plot and power spectrum as you turn the amplifier down

     • What changes do you observe when the volume is set very low?

5. Explore effects of record length and sampling rate on power spectrum

   • Play the track containing the 1 kHz sine wave

   • Set the Sampling Rate = 4000 S/s and the Record Length = 4 S

   • Make sure the power spectrum is set to Autoscale for both x and y axes

   • FIND:

     • the number of discrete points in the power spectrum (not the number of actual frequencies in the signal, but how many individual points are in the power spectrum plot)

       • Hint: the points are connected by straight line segments in the plot

     • the Frequency Resolution (= the frequency spacing between two points in the power spectrum)

     • the highest frequency in the power spectrum (not the frequency with the highest power, but the last frequency at the right side of the plot); Note: the lowest frequency in the power spectrum is always 0 Hz

       • Tip: It may help to toggle from Continuous to Hold to capture one record while you examine the power spectrum

       • Tip: If you have trouble reading off the screen - you can also save the spectrum to a file and read frequency values from it - look at Step 3 in Week 3 on how to use the Save File Options

   • Repeat the above 3 FINDs for a few longer Record Lengths (keep the number of samples low, less than 16 and always pick an even number of samples)

   • Set the Sample Rate = 8000 S/s, Record Length = 4 S and repeat the FINDs

     • Calculate the Record Duration for each case, and compare it to the frequency resolution you found for that case; are they related?

6. Observe aliasing

   • Find and play the track containing the 3 sine waves

   • Set the Sampling Rate = 5000 S/s, the Record Length = 5000 S, and toggle the switch to Continuous

   • Set the Autoscale=off switch for the x-axis on the power spectrum, and set the maximum frequency on the power spectrum axis to be 11kHz

   • Observe the 3 frequencies of the 3 peaks in the power spectrum

     • Are they the same 3 frequencies you saw when you played this track previously?

   • Increase the Sampling Rate to 7500 S/s

     • Observe any changes in the frequencies of the 3 peaks

   • Continue increasing the Sampling Rate by 2500 S/s until you get to at least 25,000 S/s

     • As you do this, observe and note down the locations of the peaks on the frequency axis

7. Examine noise

   • Set the Sampling Rate = 5000 S/s, the Record Length = 5000 S, and toggle the switch to Continuous

   • Play each of the two tracks you identified as sine waves with noise

     • What type of noise is present?

     • Estimate the signal-to-noise ratio of the signal for each track.

8. Explore the implementation of a low pass filter

   • Play the track containing the sum of three sine waves

   • Set the Sampling Rate = 22,000 S/s and Record Length = 1000 S

   • Set the Spectrum Display Settings = Amplitude-Log

   • In the time plot, Autoscale = off for the x-axis and set the maximum on the axis to 0.005 s

   • Set the Filter = on, select from the pull-down menu Low Pass, select from the next pull-down menu Butterworth, set Low Cutoff Frequency = 11,000 Hz by typing in the value

   • Paying attention to both the time plot and power spectrum of both filtered and unfiltered signals. Click the checkboxes next to "Chan 0 Filt" to turn on the plotting for the filtered signal. Keep reducing the cutoff frequency until you obtain a filtered signal that has eliminated the highest frequency sine wave

     • What cutoff frequency was required to achieve this? Why did you pick this cutoff frequency?

     • Did the filtering impact either of the other two sine waves?

9. Shutdown procedure

   • When you are through, hit the STOP APPLICATION button

   • If you are done using the DAQ please disconnect the cables from it before you transport or store it away

   • Remember the DAQ and associated peripherals are FRAGILE items!

During Week 2 lab

1. Bring your laptop, audio player if you used one (MP3 player, etc.), andDigital Sampling Kit to lab.

2. Your TA's will be grading you based on your observations, notes, and answers to the all the items in the "Prior to Week 2" section of the procedure. Make sure you take good, clean, and diligent notes so the TAs can look over your work quickly and accurately.

3. You will not lose points for incorrect inferences or answers to questions. However, you WILL lose points if the TAs deem you did not put in the appropriate amount of effort into doing the "Prior to Week 2" procedure or if you are outright missing data.

   1. As a general guideline, below is how you will be graded. Please note that there may be intangibles that affect your grade apart from what is listed here based on how your TA deems you performed your take home experiments:

   2. Full points: "Prior to week 2" data was captured diligently and notes reflect appropriate amount of effort. Most of the observations are correct and incorrect data/observations were fixed during Week 2 lab.

   3. Half points: "Prior to week 2" data was sloppy/incomplete and notes reflect inadequate effort. A non-negligible amount of the observations are incorrect, but these were fixed during Week 2 lab.

   4. Zero points: "Prior to week 2" data was not captured at all or the most of observations/notes taken were incorrect. Very little to no effort was put into completing this portion of the lab. Student is forced to finish as much of this portion of the lab as possible during week 2 lab.

4. You will have until the end of Week 2 lab to pass the in-lab assessment if any of your inferences or answers are wrong. The TAs can help you here if you are stuck, confused, or need any other help.

Prior to Week 3

After passing your Week 2 in-lab assessment, think about and decide what experiment(s) you want to perform with your DAQ/microphone system. You are required to use a microphone and DAQ but you may also use the 3.5mm audio jack breakout in your experiment if you please. You will want to find something that interests you, but also allows you to explore some interesting issues regarding the frequency content or frequency analysis of your signals. So for this part of your experiment, there are no procedures supplied.

However, as part of this experiment - like any real-world test or experiment, you will need to first validate/characterize your equipment - in this case the microphone system. The idea is to measure how well the microphone responds at various frequencies. The following procedure describes this latter process.

The following procedure is ONLY for validating your microphone. This is NOT the procedure for your experiment! That is something you must come up with and perform on your own after you have validated your microphone.

1. Setup Tone Generator

   • Find a device (computer, tablet, etc.) with a wired audio output jack

   • Locate any online Tone Generator app on the web

     • Hint: a Tone Generator will output a sine wave of a single frequency on the audio channel of your device

   • Set the volume on the Tone Generator app to at least 75% of maximum

2. Setup DAQ and Function Generator

   • Connect DAQ to the computer using the USB cable

   • Connect audio output of the device you are using to play the Tone Generator to the DAQ using the 3.5mm audio jack and jumper wires

   • Open the GTAESimpleDAQ.exe application

   • Start application (by hitting Run button)

3. Record and save single tone output

   • Use the Tone Generator to output a tone at 200 Hz

   • Set the Sampling Rate and Record Length to appropriate values

   • Set the Continuous/Hold button to Hold to capture a record of this tone

   • In the window titled Data Saving, enter a filename in the Filename text box, then click on the Power Spectrum Save button

     • Tip: the data will be saved in a folder that you had chosen/created when you started the application; also the filename will have a date-time-stamp prepended at the beginning

     • Tip: do not include a file extension in your file, it will be saved as a ".txt" file

4. Repeat Step 3 for at least 4 more tone frequencies ranging from 20 Hz to 10,000 Hz

5. Setup DAQ/microphone

   • Remove 3.5mm audio jack and jumper wires from both your device and the DAQ

   • Connect three jumper wires to the screw terminal on the microphone and to the GND, VS and AIN0 input connections on the DAQ (as instructed in the video you watched)

6. Record microphone data

   • Find a device (computer, tablet, etc.) with a good speaker system

   • Locate any online Tone Generator app on the web

     • Hint: a Tone Generator will output a sine wave of a single frequency

   • Set the volume on the Tone Generator app and on your speaker system to at least 75% of maximum

   • Use the Tone Generator to play each tone frequency from Steps 3 and 4 through your device's speakers

   • For each tone, record the microphone output with the DAQ using the same sampling setting used for that tone in Steps 3-4

   • Save the record for each tone by first entering a filename in the Filename text box (or you can use the name already there), then click on the Power Spectrum Save button

During Week 3 lab

1. Return equipment to the lab

   • Disconnect the jumper wires from the microphone and DAQ, the jumper wires from the 3.5mm audio jack

   • Carefully replace all the equipment (including wires and screwdrivers) in the original DIY kit boxes

   • Return your DIY kit boxes to the lab at your designated lab time during Week 3

   • MAKE SURE YOUR TA SIGNS YOUR LABJACK KIT BACK IN WITH THE SERIAL NUMBER. If not, you will be held responsible for it even if you have turned it in!

Data to be Taken

For "Prior to Week 2"

1. Notes taken while you were doing the required tasks (digital notes). The notes should contain the data, observations and answers to the questions posed in the Procedure section. Your notes do need to be legible, but there is no special format requirement.

For "Prior to Week 3"

1. Time histories and power spectra at 5 (or more) single frequencies (from a Tone Generator) recorded by connecting the audio output of the computer device directly to the DAQ.

2. Time histories and power spectra at the same single frequencies as in Item 1, but recorded by playing them through the device speakers and capturing them with the microphone connected to the DAQ.

3. Appropriate data, notes, observations, etc. for your own experiment.

   1. Including such topics as experimental motivation, procedure, shortcomings, problems/issues during experimentation and mitigation strategies, etc.

   2. Take GOOD notes! You WILL forget things if you do not!

​Data Reduction

For Week 3

1. From "Data to be Taken" --> "Prior to Week 3": Items 1 and 2:

   1. Calculate the relative response of the microphone-speaker system (power from microphone at each tone frequency divided by power at same tone frequency recorded from direct connection to DAQ).

2. Whatever data reduction is appropriate for your experiment.

​Results Needed for Slides Report

Note: This will be presented as an Slides Report, so you must follow instructions on how to prepare the Slides Report on the Canvas course page.

1. From your "Prior to Week 2" data/observations/notes, include a table containing waveform identification information:

   1. The track number

   2. Type of signal (ramp. product of sines, etc.)

   3. Peak principal frequency

   4. Any other peak frequencies you observed in the power spectrum

2. Any other Tables and Figures you created from "Prior to Week 2" data/observations/notes that you feel are important to present

3. Answers to any supplemental questions (None for Spring 2024)

4. From your experiment:

   1. Experimental motivation

   2. What problem you are trying to solve and/or what question you are trying to answer

   3. Experimental goals

   4. Experimental procedure

   5. Raw (if necessary) and reduced results you obtain from the experiment you designed

   6. Success criteria / conclusions

   7. Complications encountered and mitigation strategies

   8. Anything else you feel is important

      1. Think about what sort of stuff we thought was important enough for you to put into your lab reports for the previous 3 labs

   9. This is NOT an exhaustive list!

Further Reading

1. R. V. Churchill and J. W. Brown, Fourier Series and Boundary Value Problems, 3rd ed., McGraw-Hill, 1978.

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2. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed., McGraw-Hill, 1978.

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3. T. G. Beckwith, R. D. Marangoni and J. H. Lienhard V, Mechanical Measurements, 5th ed., Addison-Wesley, 1995.

   ​

4. W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flanner_, Numerical Recipes - The Art of Scientific Computing_, 2nd ed., Cambridge University Press, 1992.

Software and Files

GTAE SimpleDAQ

Navigate to the software release folder here, logging in with your GT credentials, and download the correct version for your operating system. You can also download and use the LabView source files but you will need a full LabView install on your machine for this.

Waveform files

Click to open in a new tab. Download if desired by then clicking on the ellipsis.

Objectives

The primary objective of this experiment is to familiarize the student with measurement of aerodynamic forces in a wind tunnel. A six-component load cell will be used to measure forces and moments on a rectangular (two-dimensional) wing with a removable end plate. In addition, the student will become acquainted with the operation of a subsonic wind tunnel, including use of a Pitot-static probe and pressure transducer to measure wind speed.

Background

Aerodynamics of a Wing

Forces and Moments

In general, a body moving through a fluid is subject to three force and three moment components. The primary force components on an aerodynamic body, and in particular a wing, are lift and drag. Lift is the force component that is perpendicular to the oncoming flow direction, and drag is the component parallel to the flow (see Figure 2). Lift is generally considered positive when in a direction “upward” (opposing gravity), while drag is normally positive in the flow direction (i.e., tending to slow down the body). The third component is the called the side force. For objects symmetric about the side force axis and moving straight into the flow, there should be no side force. Side forces on a wing usually come about when an aircraft is turning or not flying into the wind.

The three moments are: the pitching, roll, and yaw moments (see Figure 2). The pitching moment is around an axis parallel to the direction of the side force, and would act to change the pitch or angle of attack (a) of the lift body. The roll moment is around an axis in the drag direction, while the yaw moment acts around the lift axis. For most wings flying into the wind, the pitching moment is the dominant component. It is customary to define the pitching moment about the aerodynamic center; the point at which the pitching moment does not vary with lift coefficient, i.e., angle of attack.

The magnitudes of the aerodynamic forces and moments depend primarily on the shape of the body, the speed and orientation of the body with respect to the fluid, and certain properties of the fluid. For a rectangular wing, lift primarily depends on the curvature (camber$^1$) of its cross-section, the angle of attack, the flow speed, and the density of air. The shape of the wing’s cross-section is known as an airfoil.

The wing used in this experiment, shown in Figure 3, is symmetrical and rectangular, with a removable end plate, shown in Figure 4. Its key dimensions are given in the below table. Aerodynamics forces on the airfoil will be measured as a function of wind speed, angle of attack, and presence of the end plate.

End plates, flat plates mounted perpendicularly on the end of the wing, are "wing tip devices" commonly used in racing cars and historically used in aviation. Conceptually, these plates present a physical barrier that reduce/prevent the formation of wing tip vortices, thus improving the wing's efficiency. Raymer$^2$describes end plates as increasing the "effective aspect ratio" of the wing, leading to lower induced drag, although its presence in the flow will of course increase profile drag. Theoretically, with a decrease in span-wise flow that would otherwise have created a tip vortex, the area of the wing's surface now contributing to lift will increase.

$^1$The camber line of a wing’s cross-section connects the midway points between the upper and lower surfaces.

$^2$Raymer, Daniel P. "Aircraft Design: A Conceptual Approach", pages 266 and 298.

Aerodynamics Coefficients

Absolute aerodynamics forces on a wing are commonly non-dimensionalized and expressed as aerodynamic force coefficients. For example, the lift coefficient ($C_L$), drag coefficient ($C_D$) and pitching moment coefficient ($C_M$) are defined as:

$C_L = \frac{L}{q_\infty S}$ (1)

$C_D = \frac{D}{q_\infty S}$ (2)

$C_M = \frac{M}{q_\infty Sc}$ (3)

where $L$ is the lift force, $D$ is the drag force, $M$ is the pitching moment, $S$ is the plan form area of the wing, $c$ is the chord length (see Figure 1), and $q_\infty$is the dynamic pressure. For our rectangular wing, $S = c \times b$ , where $b$ is the span (tip-to-tip length) of the wing.

Thin airfoil theory predicts that ($C_L$) for a symmetric airfoil, i.e., where the chord and camber lines are the same (also known as a zero camber airfoil) that also has infinite span is given by

$C_L = 2\pi\alpha$ (4)

This expression is valid only for small and moderate angles of attack. At sufficiently large angles of attack, the flow over the wing no longer follows the surface, and the wing’s lift coefficient begins to drop, i.e., the wing stalls.

The dynamic pressure $q_\infty$ is given by

$q_\infty = \frac{1}{2} \rho_\infty v_\infty^2$ (5)

where $p_\infty$ is the density of the approaching flow and $v_\infty$ is its speed (relative to the body/wing). This is called the dynamic pressure because according to Bernoulli's equation (which is valid for a low speed, constant density flow) it represents the difference between the stagnation (or total) pressure and static pressure of a flow, i.e.,

$p_o - p = \frac{1}{2}\rho_\infty v_\infty^2$ (6)

The static pressure is the pressure one would measure if moving with the flow (or without requiring a change in the flow velocity), while stagnation pressure is the static pressure that the flow would achieve if it was slowed (in an ideal manner) to zero velocity. For an ideal gas, the density is given by

$\rho = \frac{p}{RT}$ (7)

Here $p$ is the (absolute) pressure, $T$ is the (absolute) temperature, and $R$ is the gas constant for the specific gas.

Sensors and Transducers

Load Cell

The device used to measure forces and moments experienced by the wing in this experiment is a six-component load cell. Being six-component, it is able to measure all three forces and all three moments. It achieves this via an array of strain gauges bonded to a machined block of hardened stainless steel. The shape and material composition of this block is cleverly designed to ensure that even complicated combinations of force and moment inputs result in distinct measurable strain outputs. Just like with the Tensile Testing lab, these strains are sensed via strain gauges, also placed strategically on the load cell body and using Wheatstone bridges to convert small changes in resistance to measurable changes in voltage. A factory calibration process relates known applied forces and moments to these voltage outputs, resulting in a mapping (called a "calibration matrix"). This calibration matrix is supplied to the customer who then uses it to transform strain gauge voltage outputs, collected via a data acquisition system, to resolved forces and moments. In our case, the load cell used is an ATI Gamma SI-130-10, which is mounted just below the wind tunnel floor, in between the wing and a stepper motor which is used to vary angle of attack. This setup, shown in Figure 5, sees the wing and stepper motor mounted vertically to eliminate the moment applied to the load cell as a result of the wing's mass. The load cell is mounted directly on the wing's quarter-chord line to remove the need to perform any offset correction. The stepper motor is pre-programmed to traverse from its initial position, having been zeroed, to an angle of attack of 20 degrees in 1 degree increments. At each angle, the stepper will dwell for 3 seconds and command the LabView VI to log the load cell measurements 1000 Samples per second.

Pitot-Static Probe

Utilizing the Bernouilli Equation (6) above, wind speed in the tunnel is measured using a differential pressure transducer, in this case a capacitance-type transducer called a Baratron. This transducer interprets the displacement of a diaphragm due to a pressure difference across the diaphragm as a change in capacitance in an electronic circuit, which is output as a DC voltage change. This voltage change is proportional to the differential pressure experienced, meaning that this voltage measured by a data acquisition system can be transformed to dynamic pressure via a simple linearly scaling. The dynamic pressure is measured directly by means of a Pitot‑static probe mounted in the freestream. The probe has one hole facing directly into the wind tunnel flow and another located so the air flows along the surface of the hole (see Figure 6). The pressure in the first hole is the stagnation pressure ( $p_o$ ) and the second experiences the static pressure ($p$). The two holes are connected via long lengths of tubing to the two sides of the Baratron shown in Figure 7, which has a known sensitivity of 1.016 mmHg/Volt.

If the data are to be meaningful, each test run must be carried out at a (nearly) constant value of wind tunnel dynamic pressure. Ideally, the magnitude of the aerodynamics force coefficients would be independent of the magnitude of the dynamic pressure. In reality, this assumption might not be completely accurate, so the dynamic pressure should be held constant during a particular data‑taking run so that the coefficients will not be influenced by changes in freestream conditions. The speed of the fan that drives the wind tunnel is nominally held constant by the fan’s motor control system. This should produce a nearly constant wind tunnel dynamic pressure. (If necessary, the dynamic pressure can be manually adjusted with the tunnel speed control.)

FlowKinetics Manometer/Flow Speed Meter

Whereas the Baratron measures only dynamic pressure, the FlowKinetics 3DP1A device also measures local ambient pressure and temperature, enabling air density and thus flow speed to be calculated. As shown in Figure 8, the FlowKinetics has a selector dial for choosing the display of either pressure or velocity (with multiple options for units on each). The FlowKinetics is an independent reference in that it does not output any signals that are read during the lab; we will be using it only for a sanity check on flow speed magnitude and for the ambient pressure and temperature measurements.

The Wind Tunnel

A wind tunnel is a duct or pipe through which air is drawn or blown. The Wright brothers designed and built a wind tunnel in 1901. The basic principle upon which the wind tunnel is based is that the forces on an airplane moving through air at a particular speed are the same as the forces on a fixed airplane with air moving past it at the same speed. Of course, the model in the wind tunnel is usually smaller than (but geometrically similar to) the full size device, so that it is necessary to know and apply the scaling laws in order to interpret the wind tunnel data in terms of a full scale vehicle. The wind tunnel used in these experiments is of the open‑return type (Figure 8). Air is drawn from the room into a large settling chamber (1) fitted with a honeycomb and several screens. The honeycomb is there to remove swirl imparted to the air by the fan. The screens break down large eddies in the flow and smooth the flow before it enters the test section. Following the settling chamber, the air accelerates through a contraction cone (2) where the area reduces (continui­ty requires that the velocity increase). The test (working) section (3) is of constant area (42" x 40"). The test section is fitted with one movable side wall so that small adjustments may be made to the area in order to account for boundary layer growth, thus keeping the stream-wise velocity and static pressure distributions constant. The air exhausts into the room and recirculates. The maximum velocity of this wind tunnel is ~35 mph, and the turbulent fluctuations in the freestream are typically less than 0.5% of the free­stream velocity. Thus, it is termed a “low turbulence wind tunnel.”

Procedure

1. Take a tour of the LTWT with the TAs, looking particularly at:

   1. The layout of the tunnel as shown in Figure 8

   2. The pitot-static probe(s) used by the FlowKinetics and Baratron

   3. The wing with removable end plate

   4. The load cell/stepper motor assembly

2. Take a safety briefing from the TAs

   1. Consult the power-on checklist for the tunnel operating procedure (also posted on the control room wall)

      1. This has changed SLIGHTLY for Fall 2024. Listen to your TAs!!

   2. Assign one or more visual observers to watch the wing during its sweeps and be ready to hit the emergency stop in the control room should anything untoward happen with the stepper motor.

   3. *READ ALL INSTRUCTIONS FULLY*

3. Help the TAs ensure everything is powered on:

   1. The wind tunnel itself (energize the master breaker near the tunnel inlet on the VFD)

   2. The ESP32 microcontroller commanding the stepper motor (USB cable should be plugged in to PC)

   3. The orange peripherals power cable (this powers the DAQ and the USB hub under the tunnel)

   4. The black emergency stop cable (this powers the stepper motor under the tunnel)

   5. The FlowKinetics manometer - follow the steps to configure it for measuring air (without calibrating the transducer or setting a correction factor other than 1.0). Make a manual note of the ambient room pressure, temperature, and air density.

   6. The large monitor - if the remote is missing use the physical button (on the underside of the front in the middle)

4. Prepare the LabView VI

   1. Open the Aero Loads VI application found inD:\AE2610\<semester year>\Aero Loads\. Drag the VI to the large TV screen so everyone can see unless the screen is already mirrored.

   2. When prompted, select a save file location in D:\AE2610Data\<semester>\<section> and an arbitrary filename (this file will not be saved).

   3. Hit the red STOP button in the top left of the VI.

   4. Select the corresponding Stepper Controller COM Port (consult Windows 10 Device Manager if required; it was COM3 as of 10/26/2024).

   5. Hit the white GO button in the top left of the VI.

   6. When prompted, select a save file location in D:\AE2610Data\<semester>\<section> and a unique filename (this file WILL be saved). Manually type in the file extension as ".csv"

   7. Verify that the Stepper Controller is successfully communicating with LabView by visually checking that Stepper Motion Status and Stepper Controller Time boxes are populated, with the time increasing. In the absence of successful comms, follow this debug sequence, re-trying after each attempt:

      1. Ensure the stepper controller is plugged in to both the PC's and controller's USB port and is powered up (red LED can be observed on the ESP32 stepper controller under the tunnel).

      2. Ensure the correct COM port is selected in the VI (double check against Windows Device Manager). Note that the VI only checks the COM port when it is first activated, any changes in the COM dropdown will be ignored after that, so you will need to stop and restart the VI for any COM changes.

      3. Power cycle the stepper controller by unplugging then re-inserting the USB cable from the PC and the orange peripherals cable. Then close and re-open the LabView VI, re-trying the opening sequence.

      4. Restart the PC, power cycle the stepper controller, and try everything again.

      5. Have a TA call Venkat Putcha

   8. Input ambient pressure and temperature from the FlowKinetics into the Air Density Calculation part of the VI. Input 287 for the Specific Gas Constant. This should yield a realistic air density (note that the graph will not populate until this is done, otherwise it will be trying (unsuccessfully) to plot NaNs.

   9. Zero the Baratron by clicking Tare Baratron.

   10. Zero the load cell by clicking Bias on the load cell control in the VI.

5. Zero the wing. Recognizing that the wing is symmetrical, and thus will produce 0 lift at 0 angle of attack, zero the yaw position:

   1. Ensure the stepper motor power is powered OFF.

   2. Utilizing an observer at the wind tunnel outlet, rotate the wing by hand until it is as close to parallel with the wind tunnel walls as possible. It will not be perfect at this time but that is not an issue as we will fine-adjust momentarily.

   3. Return to the control room and close all doors to the tunnel.

   4. Power on the stepper motor.

   5. Following the power-on checklist, get the tunnel up to full speed. Make a manual note of the wind speed. Make the following two changes to the power-on checklist for Fall 2024:

      1. Perform the initial safety test with the dial set to 20%

      2. Set max speed with the dial set to 50%. There is a physical limiter on the dial disallowing going over this power setting.

   6. Switch the LabView VI to Jog Modeand adjust the yaw position until the observed lift is 0.

   7. Click Set Home when you are happy the observed lift is 0.

   8. Power down the wind tunnel and wait for the fan to stop spinning (this may take a minute).

6. Perform the experiment

   1. Ensure the airspeed reading on the VI is close to 0 m/s. Tare the Baratron again if necessary.

   2. You will perform a total of 4 sweeps. Two different wind speeds, with and without the end plate. It doesn't matter what order you do the sweeps in so if the end plate is still installed from the previous session, do that one first, and vice versa.

   3. Power up the wind tunnel such that the speed on FlowKinetics stabilizes at 16 m/s.

      1. This will be roughly 46-48% on the dial. Keep in mind the wind speed does fluctuate so make sure the fluctuations center around 16 m/s!

      2. With Jog Mode active, command the turntable from 0 to +22 degrees in 1 degree increments. At each position, activate the logging button for a few seconds to capture load cell data.

      3. Note down and discuss anything interesting that you see or capture with your group and TA!

   4. Return the wing to its zero position by clicking Go Home. If the loads are not zero after returning home, you may need to re-zero the angle of attack and set a new home position as you did in "Step 5.5-5.7: Zero the Wing" above

   5. Decrease the wind speed of the tunnel such that the FlowKinetics reading stabilizes at 12 m/s

      1. This will be roughly 34-37% on the dial. Keep in mind the wind speed does fluctuate so make sure the fluctuations center around 12 m/s!

      2. With Jog Mode active, command the turntable from 0 to +20 degrees in 1 degree increments. At each position, activate the logging button for a few seconds to capture load cell data.

      3. Note down and discuss anything interesting that you see or capture with your group and TA!

   6. Power down the wind tunnel. Wait for the fan blades to stop spinning before exiting the control room (this may take a few minutes)

   7. Open the tunnel side panel. If the end plate is installed, remove it. If the end plate is not installed, install it. This is achieved using the provided hex key and the 6 screws.

   8. Zero the load cell by clicking Bias on the load cell control.

   9. Return the tunnel to full speed, following the power-on checklist. Keep the following changes in mind:

      1. Perform the initial safety test with the dial set to 20%

      2. Set max speed with the dial set to 50%. There is a physical limiter on the dial disallowing going over this power setting.

   10. Zero the wing again using the procedure above if necessary

   11. Repeat Step 6.3-6.5 above to capture 2 new data sets with this new configuration.

   12. Power down the wind tunnel when you have finished gathering all 4 data sets.

7. Data check

   1. Stop the VI.

   2. Open the file you selected as your save file in MS Excel. If you didn't give it a file extension when you selected the save file you will need to add it now by right-clicking on the file, clicking Rename, and then adding ".xls" to the end of the filename.

   3. Generate quick scatter plots of Fx, Fy, Tz, and Baratron Voltage against Angle-of-attack to sanity check your logged data. If any data is missing or looks wildly corrupt, repeat the relevant sweeps.

   4. Once successful data has been acquired, ensure it is emailed to your group or uploaded to Canvas before leaving the lab.

8. Assist the TAs in shutting down the experiment

   1. Power down the tunnel master breaker

   2. Unplug the ESP32 stepper controller USB cable from the PC

   3. De-energize the stepper motor and peripherals by flipping the switch on the power strip in the control room

   4. Turn off FlowKinetics

Data to be Taken

1. From the FlowKinetics device: ambient air density, temperature, and pressure.

2. Tared forces, moments, and baratron voltage of the wing at 20/22 angles of attack and two wind tunnel speeds, with the end plate.

3. Tared forces, moments, and baratron voltage of the wing at 20/22 angles of attack and two wind tunnel speeds, without the end plate.

Data Reduction

1. For each batch of force/moment and Baratron voltage data collected at each angle of attack, calculate its mean and variance.

2. Convert the mean forces/moments in the balance reference frame to the wind frame (i.e., convert to lift and drag). To do this you will need to know the following facts:

   1. The wing's centerline is perfectly aligned along the y-axis of the load cell, with the front of the wing being in the negative-y direction

   2. The load cell's y-axis is oriented such that positive-y is pointing down the wind tunnel, and the x-axis is perpendicular to the y-axis, with positive-x pointing towards the control room

   3. The origin of the load cell is located on the quarter-chord line of the wing

3. Convert the mean Baratron voltage to wind speed using the known relationship with dynamic pressure discussed above, and the manually recorded ambient air density.

4. Express all forces and moments in non-dimensionalized coefficient form. The chord to be used is the chord of the wing from the wing leading edge to the flap trailing edge as shown in Table 1 above, and the dynamic pressure to be used is that calculated in the previous step.

5. Uncertainty Information needed for Uncertainty Analysis:

   1. Baratron Voltage: +/-0.002 V

   2. Load: +/-0.05 N

   3. Torque: +/-0.001 N-m

Results Needed for FORMAL Report (also see instructions on Canvas)

1. A table containing wing geometry ‑ chord of wing, span, planform, thickness ratio.

   ​

2. Plot a figure showing lift coefficient (ordinate) versus angle of attack (CL vs. $\alpha$) for the wing at the two wind speeds, with and without end plates.

   ​

3. Plot a figure showing drag coefficient (ordinate) versus angle of attack (CD vs. $\alpha$) for the wing at the two wind speeds, with and without end plates.

   ​

4. Plot a figure with drag coefficient as abscissa and lift coeffi­cient as ordinate (CL vs. CD – also known as a drag polar) for the two wind speeds, with and without end plates.

   ​

5. Plot a figure with moment coefficient (ordinate) versus angle of attack (CM vs. $\alpha$) for the wing at the two wind speeds, with and without end plates.

Objectives

This experiment is intended to explore the stress-strain behavior of ductile metals subject to tensile loads and to introduce transducers that are used in mechanical testing. The tensile test is the most basic structural test of a material and is used to characterize its response to structural loads. The characterization data obtained from a tensile test is used directly for structural analysis and design. This experiment requires the use of several mechanical transducers that are used to measure the elongation of an aluminum specimen and the force applied by a load frame. As part of this lab, you will also be exposed to the physical principles that are used in these devices. Two tests will be performed: (1) a monotonic loading where the specimen will be gradually loaded until failure to observe the full stress-strain response, (2) a cyclic loading where a second specimen will be repeatedly loaded and unloaded until failure to observe elastic versus inelastic behavior.

Safety

Wear close-toed shoes in the lab to avoid injuries to your feet if you drop lab equipment, and the eye protection glasses supplied to you. This experiment also involves the use of chemical adhesives and solvents; so make sure to wash your hands after performing the lab.

Background

Stress and Strain

A tensile test is designed to experimentally characterize the relationship between stress and strain. Stress and strain are fundamental concepts in the study of mechanics of materials and we briefly summarize them here.

Stress is a measure of the intensity of a force exerted over an area. In this test, we will measure axial or normal stress. Consider a prismatic bar with a cross-sectional area , loaded at both its ends with opposing forces with magnitude P, the stress in the bar is given by:

(1)

which is the normal stress. Stress has units of force per unit area, which in SI units are pascals [Pa] and in English units are pounds per square inch [psi]. Since the magnitude of stress can be large, it is common to use units of megapascals [MPa] and ksi (i.e., thousands of psi). Note that during the tensile test, the cross-sectional area of the specimen will change; however, we will continue to normalize the force by the initial area. This is usually referred to as engineering stress.

(2)

Tensile Testing Specimens

In this experiment, you will measure the strain in a bar using a strain gauge, and the elongation of the bar under load using an extensometer. These measurement devices are described in a later section.

You will use aluminum that has been machined into a typical material specimen shape, specifically, a dog bone specimen, as shown in Figure 1. The shape of the specimen is designed to produce a uniform tensile stress and tensile strain in the gauge section. If the specimen were to fail outside this region, in either the grip or shoulder areas, then the results from the test could not be used because stress and strain in these regions are not uniform. The dog bone specimen is designed so that we can fasten the test machine to the grips at either end of the specimen to apply an axial load.

Stress-Strain Diagrams and Material Models

In this experiment, you will generate experimental stress-strain diagrams. A sketch of a stress-strain diagram for a generic ductile metal is shown in Figure 2. As engineers, we use mathematical approximations that are designed to model the measured stress-strain response. Thus, we can explore how closely these mathematical models match the measured stress-strain diagram.

Material models can be classified as either elastic or inelastic. An elastic response is one in which the loads on the structure are low enough so that no permanent deformation or damage to the specimen occurs. Once the loads are removed, the specimen returns to its original shape, and the experiment could be repeated to produce the same stress-strain response. An inelastic response is one in which permanent deformation occurs, so that when the specimen is unloaded it returns to a different shape than the original specimen.

The simplest mathematical model of an elastic stress-strain response is Hooke's Law. Hooke postulated a linear relationship between stress and strain, written as follows:

(3)

True Stress and Strain

Thus far, we have examined engineering stress and engineering strain, which are force and elongation normalized by the INITIAL geometric properties. In addition, there are also true stress and true strain; they represent the force normalized by the instantaneous area, and the elongation normalized by the instantaneous length.

By combining the area ratio above with the expression for engineering stress, Eq. (1), the true stress is given by:

(4)

Similarly the true strain can be related to the engineering strain,

(5)

Within the strain hardening regime, ductile materials often satisfy the strain hardening law:

(6)

where K is called the strength coefficient, and n is called the strain hardening exponent. On a logarithmic scale, this strain hardening law becomes linear:

(7)

Plotting the logarithm of the true stress and true strain for the entire loading history produces a piecewise linear curve, where the first curve is from the linear elastic regime, that is,

(8)

Tensile Testing Equipment

Load Frame

In this lab, we will use a load frame to apply a tensile load to a coupon of an aluminum alloy. The load frame in our lab is an Instron 5982 (see Figure 4) that is capable of delivering 100 kN of axial force to the specimen for testing purposes. It consists of two columns with a crosshead and a base. The crosshead moves up and down the columns driven by lead screws. The crosshead is attached to a load cell, which measures the force applied to the specimen. The test specimen is fastened between two test fixtures attached through pin joints to the base of the frame and the load cell/crosshead. The pin joints ensure that no moments are transmitted to the test specimen.

The load frame controller can operate in either displacement-control mode or a force control mode. In load control mode, the load frame applies a commanded load and measures the extension. Under displacement control, the controller measures the applied force through the load cell and finds the force required to produce the specified displacement. For this experiment, you will use displacement control to measure the full response of the specimen up to the point of rupture or failure.

Strain Gauge

A strain gauge is a device that is used to measure the strain at the surface of a structure. The gauge consists of a thin conductive metal foil grid pattern that is mounted on a flexible backing (see Figure 5).

(9)

Before beginning the structural test, you will first bond the pre-wired linear strain gauge to the specimen. The quality of the bond between the strain gauge and specimen has a direct impact on the quality of the strain measurements. Achieving a good bond between the strain gauge and the specimen requires careful surface preparation while avoiding contamination. Surfaces that have not been thoroughly cleaned must be treated as if they are contaminated. Touching the strain gauge contaminates it and can lead to poor bond quality. Therefore before bonding the strain gauge to the specimen, the surface of the specimen must be prepared.

Extensometer

The extensometer is a linear variable differential transformer (LVDT) that measures the relative displacement between two points on the specimen. As shown in Figure 6, the LVDT is attached to the specimen using arms that are attached to the body of the extensometer. In the lab, you must measure the initial distance between the attachment points on the specimen.

The LVDT is a transducer that measures the displacement using the principle of induction. The LVDT consists of a central primary coil and two secondary coils wired in sequence within an assembly with a movable internal core. An AC current is passed through the primary coil while the induced current is measured in the secondary coils. The secondary coils are wound in opposite directions on either side of the primary coil. The movable core within the assembly is attached to the object whose displacement is being measured. An AC current in the primary coil induces an AC current in the second coil that depends on the relative displacement of the core. The displacement of the core within the LVDT can be estimated by measuring the output AC signal relative to the input signal. An advantage of the LVDT is that it experiences little wear during use and is very robust. In addition, the LVDT can have a high resolution of the displacement.

Load Cell

A load cell is a transducer that is used to measure force. Most load cells are made from an arrangement of strain gauges on a load carrying material with known material properties. The strain gauges are arranged to reduce the sensitivity of the measurement. In subsequent labs, we will see a load cell used to measure forces on an airfoil in a wind tunnel. These instruments and transducers for measuring the stress-strain data are connected as shown in Figure 7.

Pre-Lab Preparation

Before coming to lab:

1. Read this lab manual thoroughly

2. Watch the following online Vishay Precision Group videos about surface preparation and strain gauge bonding. (Not Needed Fall 2024-Present)
3. Read the MicroMeasurements strain gauge application technical bulletin below. (Not Needed Fall 2024-Present)

Procedure

Specimen Preparation

1. Gather your materials

   1. Obtain 2 dogbone test specimens from the TAs.

      1. One of these will be Aluminum 2024-T3

      2. The other will be a "mystery material" specimen

2. Determine the test specimen dimensions

   1. Using the ruler and calipers provided, measure the length, width, and thickness of the reduced section of the test specimen.

      1. The reduced section is the section of the specimen that is narrower

         1. Do NOT include the transition region of the specimen (the region where the cross sectional area is getting smaller). See the figure below for reference.

         3. Have multiple members of your group perform this measurement so you can compare and converge on the correct dimensions.

   3. Note: The dog bone specimens are manufactured in the AE Machine Shop using the CNC (computer numerical control) waterjet cutter and geometry defined by a CAD (computer aided design) file. The dimensions can be determined from the CAD file directly. However, machining imperfections can result in varying geometry so it's always best to verify accurate dimensions post-fabrication.

Monotonic Loading Test

In this test, continuous load and strain data will be acquired by the system while it automatically loads a test specimen in tension until it fails. Load will be applied to the specimen by traversing the crosshead upwards at a rate of 0.15 inches per minute. Strain data will be acquired from an extensometer that is clipped onto the test specimen.

1. Prepare the Instron - To operate the Instron it must be powered on and have a successful connection with the Bluehill software on the PC. To do this:

   1. Turn on the main power switch on the right hand side of the base of the frame; it will take a minute or so to fully initialize.

   2. Turn on the PC connected to the Instron load frame; small button on the back of the PC.

   3. Open the Bluehill Universal software. Once it is open, the Instron Machine will connect to the PC (you will hear a click/pop) and you'll see the control panel on the Instron displaying position/load values.

2. Open the appropriate testing method - Go to Bluehill and configure the following items:

   1. In Bluehill, Click Test and select Browse Methods under "New Sample"

   2. Navigate to the current folder: C:\Users\Public\Documents\Instron\Bluehill Universal\Templates\AE_Labs\AE2610\<Semester Year> and open the "AE2610_Monotonic.im_tens" method

   3. Once the sample is ready for testing, select OK. There is no need to set any travel limits as they have already been set for you.

      1. If you see any errors thrown during this part of the process, there is a method error or a connection error. If an error persists, call the Head TA or the Lab Manager.

      2. DO NOT edit the method or fix Instron connections on your own!

   4. Your screen should appear as follows:

3. 2. On the extensometer, there are two knobs (one on each arm of the extensometer). One knob has a cone and the other has a dimple. Gently press the cone into the dimple such that the arms of the extensometer do not move. What this process does is ensure that you are now holding the knife edges exactly 2 inches apart, which is the gauge length of this particular extensometer. See the images below for reference.
   3. Input the parameters as follows:

      1. Gauge Length: 2 in

      2. Calibration type = Automatic

      3. Click Calibrate and OK

      4. Once calibration is complete, click Close

   4. Close the System details window if the transducer calibrated properly

      1. If the calibration fails, check the wiring - usually it fails when there is an open circuit which can come from a faulty connection

      2. Note: If the transducer icon turns gray, it means that the transducer did not calibrate properly

4. Install the test specimen in the Instron - With the assistance of your TAs, carefully mount, align, and clamp your test specimen into the the grips of the Instron.

   1. If the Tensile Grips are not already installed in the machine, aid your TA in mounting the Grip attachments to the Instron load frame.

   2. Traverse the crosshead up/down to the needed crosshead height. It should be at a height at which you can grip both ends of the dogbone specimen.

      1. Note: Be careful when jogging the Instron up/down. Take care not to pinch your fingers or bend the specimen. The crosshead applies A LOT of load with very little displacement! Use the fine position dial if needed.

   3. Mount your specimen. Ensure that your specimen is square to the jaws of the instron. If it is not, you will be loading the specimen at an angle.

   4. Tighten both grips such that they are gripping the dogbone specimen securely.

5. Install the extensometer

   1. Verify the extensometer is working correctly by observing the live display of extensometer strain while carefully moving the knife edges manually. If there are any issues, have the TAs call the Head TA or the Lab Manager to help debug.

   2. On the extensometer, gently press the cone into the dimple such that the arms of the extensometer do not move (same process as above).

   3. Mount the extensometer in the center of the test specimen's necked region ensuring that the knife edges are not moved from their 2 inch gauge length.

   4. If either knife edge of the extensometer is loose and does not grip to the dogbone specimen, inform your TA. They will fix this.

6. Run the test

   1. Ensure everyone and everything is clear of the Instron and everyone is wearing the appropriate PPE. Ensure that the blast door is closed.

   2. On the bottom of the screen, click Balance All and Zero Displacement- this zeroes all values. Observe the 4 live readings at the top of the screen to ensure everything is essentially zero.

      1. If you start to see drift in your extensometer reading, ensure that the knife edges are not sliding down the neck of the dogbone specimen; You may need to slightly adjust where the extensometer is clamped. If the issue persists, call the Head TA or Lab Manager.

   3. When you are ready, hit Start.

   4. During the test, observe the force-strain plots for the extensometer (bottom plot) as the Instron's crosshead traverses upwards. Discuss anything of interest that you see occurring.

   5. Note: If at any point during the test you or the TAs see something going wrong, click Stop in Bluehill. If this does not work, hit the E-Stop button on the right pillar of the Instron Machine.

7. At the end of the test

   1. Once the specimen fails, Bluehill will automatically stop the crosshead.

   3. Verify the data output file - In Windows Explorer, navigate to C:\Users\Public\Documents\Instron\Bluehill Universal\Templates\AE_Labs\AE2610\<Semester Year>

      1. Open the "Monotonic-1.csv" file containing your raw data in Notepad

      2. Ensure your data is present and accounted for; scroll through the data and look at the numbers to make sure all the data is present and accurate. Don't simply open and close the file!

      3. Close the csv file and rename it to "Monotonic_Axx" with "xx" being your section number

      4. Open the "Section Data" Folder and create a new folder with your section number (e.g. A14). Cut and paste the "Monotonic_Axx.csv" file into your sections folder

   4. Remove the extensometer from the broken test specimen and remove the specimen from the Instron jaws.

   5. Measure the fractured region area - Using the digital caliper provided, measure the cross-sectional dimensions of the specimen's fractured region--the specimen width and thickness exactly where it has fractured.

   6. Measure the uniform deformation region area - Using the digital caliper provided, measure the width and thickness of the region adjacent to the final fracture that undergoes uniform deformation (about 1cm away from where it has fractured is a good location to measure this).

   7. Measure the final length of the specimen - Place the two parts of the specimen roughly together. Using the ruler provided, measure the length of the reduced section of the specimen.

   8. Repeat the experiment if needed - check and discuss your data with the TAs to ensure that the experiment proceeded as expected. If it did not, you may need to run the experiment again.
8. REPEAT THE ENTIRE SPECIMEN PREPARATION AND MONOTONIC TEST PROCEDURE WITH THE MYSTERY SPECIMEN

Data To Be Taken

1. Reduced section dimensions of both specimens:

   1. Length, width, and thickness of the specimen before testing

   2. Width and thickness of the fracture region after failure

   3. Width and thickness of the uniformly deformed region after failure

   4. Final length after failure

3. Two raw csv data files (one for each of the two) Monotonic Specimens containing:

   1. Time

   2. Load

   3. Strain Gauge Strain (this data will be recorded, but will not be used as of Fall 2024-Present)

   4. Extensometer Strain

Data Reduction (Do this for both specimens!)

1. Compute the engineering stress at each data obtained point based on the load data from the Instron and the initial cross-section area you measured

2. Using the linear elastic region:

   1. Isolate and truncate a segment of the data from this region from each data set

   2. Estimate the Young’s modulus

3. Using the load and strain data, estimate the following for both specimens:

   1. Engineering ultimate tensile strength (UTS), in [ksi] and [MPa] from the initial specimen dimensions

   2. True ultimate tensile strength, in [ksi] and [MPa] from the uniform deformation region dimensions

   3. True fracture stress, (in [ksi] and [MPa]) from your fractured specimen dimensions

   4. Engineering fracture stress, (in [ksi] and [MPa]) from the initial specimen dimensions

   5. True ultimate tensile strain from extensometer readings

   6. Engineering fracture strain from your initial and fractured specimen dimensions

   7. True fracture strain from your initial and fractured specimen dimensions

   8. True fracture strain from extensometer readings

4. From the data obtained in the plastic regime, use the procedure explained in the background section to determine the strain hardening index n and strength coefficient K

Results Needed For Data Report

Note: Closely follow the instructions on Canvas for preparation of the Data Report.

1. A table containing the following values for both specimens:

   1. Reduced section length, width, and thickness before testing

   2. Fracture region width and thickness after failure

   3. Uniform deformation region width and thickness after failure

   4. Final length of reduced section after failure

   5. Gauge length of extensometer

2. A single graph of engineering stress vs. strain for the aluminum specimen

3. A single graph of engineering stress vs. strain for the mystery specimen

4. A table that lists three elastic (Young’s) modulus in both ksi and MPa units

   1. Young's modulus estimate of the aluminum specimen

   2. Young's modulus estimate of the mystery specimen

   3. Published Young's modulus values for the aluminum alloy used in the experiment, which is 2024-T3. You will be responsible for finding a reputable source for this published value!

5. Two tables (one for each specimen) containing:

   1. Engineering ultimate tensile stress in ksi and MPa

   2. True ultimate tensile stress in ksi and MPa

   3. Engineering fracture stress in ksi and MPa

   4. True fracture stress in ksi and MPa

   5. True ultimate tensile strain obtained from extensometer

   6. True fracture strain from extensometer

   7. True fracture strain calculated from dimensions

   8. Engineering fracture strain calculated from dimensions

6. Two graphs (one for each specimen) of the stress vs. extensometer strain for the monotonic specimen with two curves:

   1. One curve with the the engineering stress

   2. A second curve with the true stress

      1. Plot the true fracture stress and true ultimate tensile stress points calculated above on the same plot. Label these points clearly!

      2. From the start of the plastic region, roughly try to connect the true stress-strain points to create a true stress-strain plot. Try to look up some literature to do this properly!

   3. Clearly label the plots in such a way to distinguish between the engineering and true stress curves
7. A table of your measured strain hardening index n and strength coefficient K for both specimens

8. A table with your guess as to what the mystery material is and a justification for your selection. Your options include:

   1. 145 Copper

   2. 4130 Steel

   3. 220 Nickel

   4. 110 Copper

   5. 510 Bronze

   6. Stainless Steel 316

   7. 260 Brass

   8. Aluminum 7075

   9. 353 Brass

   10. 17-4 Stainless Steel

   11. Inconel 625